Floresta da Tijuca
Floresta da Tijuca is the largest urban forest in the world, and is my favorite escape from the city. In my opinion, some of the the most beautiful spots Rio has to offer are located inside this forest. There are several exciting peaks to visit that have magnificent 360 degree views of the city. There are also numerous waterfalls located throughout the forest that are great for cooling off, as well as some incredible caves that are hidden out of site of the better marked trails. It boasts a wide range of flora and fauna and it is quite common to see monkeys, birds, and all sorts of other indigenous wildlife.
Due to its size, I generally suggest full day treks throughout the forest, and the difficulty level can range from quite simple to more strenuous. I have my preferred routes that I feel show the wide variety of this forest (all which should be easily accomplished by any well fit individual), but with so many options, it is easy to personalize a trek that is just right for you.